2.1.1. Coupling between the free-surface sloshing problem and the lateral vehicle motions

The free-surface sloshing problem by Faltinsen et al. (Reference Faltinsen, Rognebakke, Lukovsky and Timokha2000, (2.2)) is normally formulated in the tank-fixed coordinate system $Oyz$. When the translational velocity

(2.1a,b)\begin{equation} {\boldsymbol v}_O(t)=b \left(\dot\eta_2(t),0\right),\quad \eta_{2b}(t)=b\eta_2(t) \end{equation}

is determined by the non-dimensional generalised coordinate (sway) $\eta _2(t)$, adopting the spatial normalisation by $b$ and introducing the $b$-scaled relative velocity potential,

(2.2)\begin{equation} \phi(y,z,t)=\varPhi(y,z,t)/b^2-y\dot\eta_2(t), \end{equation}

where $y$ and $z$ are the non-dimensional body-fixed coordinates, transform the two-dimensional free-surface sloshing problem to the form

(2.3)\begin{equation} \left. \begin{gathered} \nabla^2\phi=0\quad \text{in}\ Q(t);\quad \frac{\partial\phi}{\partial n} =0 \quad \text{on}\ S(t),\\ \frac{\partial\phi}{\partial n} =\frac{\partial\zeta}{\partial t}/\sqrt{1+({\partial\zeta}/{\partial y})^2}\quad \text{on}\ \varSigma(t); \quad \int_{{-}l/2}^{l/2}\zeta\,\textrm{d}y =0,\\ \frac{\partial\phi}{\partial t}+ \frac 12 (\boldsymbol{\nabla}\phi)^2 + \underbrace{\bar g}_{g/b}\zeta ={-}y\ddot\eta_2(t)\quad \text{on}\ \varSigma(t), \end{gathered} \right\} \end{equation}

where $g$ is the gravity acceleration, $z=\zeta (y,t)$ defines the free surface $\varSigma (t)$, $Q(t)$ is the time-dependent liquid domain, $S(t)$ is the wetted tank surface, which are normalised by $b$, and ${\boldsymbol n}$ is the outer normal.

Because $\ddot \eta _2(t)$ in the dynamic boundary condition of (2.3) is unknown, one should, in addition, introduce the Newton law with respect to $\eta _2(t)$,

(2.4)\begin{equation} b (M_t+M_l)\,\ddot \eta_2 = {\mathcal{F}}_{ext}(t)+ {\mathcal{F}}_{slosh}(t), \end{equation}

where $M_t$ is the rigid tank mass and $M_l=\rho _l V_l$ is the liquid mass ($\rho _l$ and $V_l$ are the liquid density and volume, respectively). The present section excludes from consideration both the frictional (structural damping) and restoring forces. As for the restoring force, it is negligibly small for sway motions of a floating tank, whose studies are the primary goal of the present paper. No damping in the original formulation is a mathematical requirement of the multimodal and almost all analytical methods in sloshing problems. The methods need the linear eigensolution (natural sloshing modes and frequencies) of the (here, coupled tank-slosh) problem, which becomes mathematically impossible with non-zero damping in the mechanical system. However, the damping terms can be incorporated into the modal equations after these are derived from the undamped formulation. How to do that is demonstrated in § 3.

The sloshing-related horizontal hydrodynamic force ${\mathcal {F}}_{slosh}(t)$ should be derived from a pressure integral over the wetted vertical tank walls, where the pressure is computed by using the Bernoulli equation in the body-fixed coordinate system (Faltinsen & Timokha Reference Faltinsen and Timokha2009, (2.60)). Alternatively, ${\mathcal {F}}_{slosh}(t)$ can be found by using the Lukovsky formula (see details in Faltinsen & Timokha (Reference Faltinsen and Timokha2009), chapter 7),

(2.5)\begin{equation} {\mathcal{F}}_{slosh}(t)={-} M_l \ddot y_C(t)={-} M_l \frac{\textrm{d}^2}{\textrm{d}t^2} \int_{Q(t)}y\, \textrm{d}Q/V_l={-}\frac{M_l}{b^2h} \int_{{-}1/2}^{1/2}y\, \frac{\partial^2}{\partial t^2}\zeta(y,t)\,\textrm{d}y, \end{equation}

which analytically expresses the horizontal hydrodynamic force in terms of the horizontal coordinate $y_C(t)$ of the liquid mass centre.

Equations (2.3)–(2.5) govern the coupled sloshing-tank dynamics, where the three $b$-normalised unknowns $\zeta (y,t)$, $\phi (y,z,t)$ and $\eta _2(t)$ are defined in the body-fixed coordinate system $Oyz$. The unknowns are fully coupled, i.e. $\eta _2$ is present in the free-surface problem (2.3) and $\zeta$ appears in the Newton law (2.4).

2.1.2. Decoupling the free-surface sloshing problem from $\eta _2(t)$

Substituting the Lukovsky formula, (2.5), into (2.4) gives

(2.6)\begin{equation} \ddot \eta_2(t) = \underbrace{\frac{{\mathcal{F}}_{ext}(t)}{b(M_t+M_l)}}_{=\bar{\mathcal{F}}_{ext}(t)} - \underbrace{\frac{M_l}{(M_l+M_t)h}}_{{=}K>0} \int_{{-}1/2}^{1/2}y\, \frac{\partial^2}{\partial t^2}\zeta(y,t)\,\textrm{d}y= \bar{\mathcal{F}}_{ext}(t)-K\ddot y_C(t). \end{equation}

Furthermore, using this formula in (2.3) partly decouples (2.3)–(2.5) so that (2.3) takes the form

(2.7)\begin{equation} \left. \begin{gathered} \boldsymbol{\nabla}^2\phi=0\quad \text{in}\ Q(t);\quad \frac{\partial\phi}{\partial n} = 0 \quad \text{on}\ S(t),\\ \frac{\partial\phi}{\partial n} =\frac{\partial\zeta}{\partial t}\left/\sqrt{1+\left(\frac{\partial\zeta}{\partial y}\right)^2}\right.\quad \text{on}\ \varSigma(t); \quad \int_{{-}1/2}^{1/2}\zeta\, \textrm{d}y=0,\\ \frac{\partial\phi}{\partial t}+ \frac 12 (\boldsymbol{\nabla}\phi)^2 -\underline{K y \int_{{-}1/2}^{1/2} y \frac{\partial^2\zeta}{\partial t^2}\,\textrm{d}y}+ \bar g \zeta ={-}y\, \bar{\mathcal{F}}_{ext}(t)\quad \text{on}\ \varSigma(t), \end{gathered} \right\} \end{equation}

where $K$ and $\bar {\mathcal {F}}_{ext}(t)$ are defined in (2.6). The free-surface (sloshing) problem (2.7) does not contain the tank-related generalised coordinate $\eta _2(t)$; it exclusively links $\zeta (y,t)$ and $\phi (y,z,t)$. The problem describes sloshing visible through a camera installed on the rigid body .

The two mathematical formulations (2.3)–(2.5) and (2.6)–(2.7) are equivalent. They describe the coupled slosh-vehicle dynamics, forced (${\mathcal {F}}_{ext}\not =0$) and/or unforced (${\mathcal {F}}_{ext}=0$). The coupled (linear) eigenfrequencies following from these mathematical formulations were, for instance, computed by Herczyński & Weidman (Reference Herczyński and Weidman2012). The linear frequency-domain problem can also be solved by employing the sloshing-related frequency-dependent sway added-mass coefficient $A_{22}^{slosh}(\sigma )$ from Faltinsen & Timokha (Reference Faltinsen and Timokha2009, (5.134)) that deduces the governing equation

(2.8)\begin{equation} \left(M_t+M_l+A_{22}^{slosh}(\sigma)\right)\ddot\eta_{2b}(t)={\mathcal{F}}_{ext}(t) \end{equation}

from the Newton law (2.4) so that the dispersion equation for computing the coupled eigenfrequencies implies the zero coefficient at $\ddot \eta _2(t)$, i.e. $M_t+M_l+A_{22}^{slosh}(\sigma )=0$. Here, because $A_{22}^{slosh}(\sigma )\to \pm \infty$ as $\sigma \to \sigma _1\pm$, the solution $\sigma _{s,1}>\sigma _1$.

The nonlinearity of the coupled sloshing-tank dynamics is exclusively associated with the free-surface problems (2.3) and (2.7). When considering the prescribed tank motions ($\eta _2(t)$ is known in (2.3)), one can construct analytical approximations of the nonlinear resonant sloshing problem (2.3) in terms of the Stokes standing-wave (natural sloshing) modes in the stationary two-dimensional rectangular tank (Faltinsen & Timokha (Reference Faltinsen and Timokha2009), chapter 4). An example is the Narimanov–Moiseev multimodal theory by Faltinsen et al. (Reference Faltinsen, Rognebakke, Lukovsky and Timokha2000). This and other asymptotic analytical theories require that the forcing frequency $\sigma$ is close to the first natural sloshing frequency $\sigma _1$. The closeness is the necessary condition. However, the resonance coupled eigenfrequency $\sigma _{s,1}\not =\sigma _1$ and, therefore, applying the theories to (2.3) does not guarantee they provide an accurate prediction of the resonant coupled motions.

Because the free-surface problem (2.7) is self-contained and determines the main resonant properties of the coupled tank-sloshing mechanical system ((2.6) simply returns $\eta _2(t)$ for the given wave elevations by $\zeta (y,t)$), the coupled eigenfrequencies should be a subset of the natural sloshing frequencies following from the linearised and unforced (2.7). Moreover, the boundary value problems (2.3) and (2.7) are mathematically similar and differ only by the underlined integral term in (2.7) (in the dynamic boundary condition on $\varSigma (t)$), which is absent in (2.3). Hence, as long as we know an analytical approach to the free-surface problem (2.3) with the prescribed acceleration $\ddot \eta _2(t)$, where $\sigma$ is close to the lowest natural sloshing frequency $\sigma _1$, mathematically, the approach could be extended to the free-surface problem (2.7) with the prescribed force $\bar {\mathcal {F}}_{ext}(t)$, where $\sigma$ is close to $\sigma _{s,1}$. How to do this extension for the nonlinear multimodal approach by Faltinsen et al. (Reference Faltinsen, Rognebakke, Lukovsky and Timokha2000) will be described in the next subsections. The procedure suggests constructing the corresponding non-Stokes natural sloshing modes and frequencies, derivation of the Narimanov–Moiseev-type (single-dominant) modal system, and studying its periodic solutions, which describe the steady-state resonant sloshing coupled with lateral tank motions.

2.2.1. Frequencies $\sigma _{s,m}$

The dispersion equation (2.15) expresses balance between inertial forces of the rigid tank + frozen liquid mechanical system and the hydrodynamic sloshing forces. In the limit $N\to \infty$, it has an infinite set of positive real roots $\{\sigma _{s,2k-1}^2,\, k\ge 1\}$, which satisfy

(2.16)\begin{align} \sigma^2_{2i-1}<\sigma^2_{s,2i-1}<\sigma^2_{2i+1},\quad i=1,2,\ldots \quad \text{and}\quad \sigma^2_{s,2i-1}/\sigma^2_{2i-1}\to 1+\quad \text{as} \ i\to \infty. \end{align}

Because the even (symmetric) sloshing modes are not coupled with lateral tank motions in the linear approximation,

(2.17)\begin{equation} \sigma^2_{2i}=\sigma^2_{s,2i}, \quad i=1,2,\ldots . \end{equation}

However, the symmetric sloshing modes play an important role in the nonlinear resonant sloshing theories.

The theoretical ratios $\sigma _{s,1}/\sigma _1$ and $\sigma _{s,3}/\sigma _3$ between the non-Stokes and Stokes natural sloshing frequencies are shown in figure 2(a) versus $M_t/M_l$ for three finite liquid depths $h$. The figure illustrates that the ratios tend to one with increasing $M_t/M_l$ (heavyweight tank) and $h$ (deep water). Herczyński & Weidman (Reference Herczyński and Weidman2012) have experimentally investigated $\sigma _{s,1}$ and $\sigma _{s,3}$ versus $M_l/M_t$ by varying the tank filling and compared them with the theoretical values by (2.15). Because $h$ and $M_l/M_t=[M_l/M_t](h)$ changes in a complex way in these experiments, we were not able, technically, to incorporate the measured values in figure 2(a). On the other hand, comparison by Herczyński & Weidman (Reference Herczyński and Weidman2012) confirms a good agreement between the theoretical formula (2.15) and the experimental data.

Figure 2.

Natural sloshing frequencies and modes for the two-dimensional coupled sloshing-rectangular tank motion problem illustrated in figure 1(a). Panel (a) shows the two lowest antisymmetric non-Stokes natural sloshing frequencies normalised by the corresponding Stokes natural sloshing frequencies, $\sigma _{s,1}/\sigma _1$ (the solid lines) and $\sigma _{s,3}/\sigma _3$ (the dashed deep-blue lines) versus the tank-liquid mass ratio $M_t/M_l$ for three non-dimensional liquid depths $h=0.3, 0.5$ and 0.7 (the $h$-values are used as the curves labels). Panels (b,c) compare the non-Stokes ($z=f_{s,1}(y)$ and $=f_{s,3}(y)$ by (2.22), the deep-blue dashed lines) and Stokes ($z=f_1(y)$ and $=f_3(y)$, the bold black solid lines) standing wave profiles for different $M_tM_l$ and $h$. Panel (b) focuses on the weightless tank ($M_t/M_l=0$); the curves are marked by the $h$ values. The mean liquid depth $h=0.5$ is adopted in panel (c), the curves are tagged by the $M_t/M_l$ values.

2.2.2. Modes $\varphi _{s,m}$

Because of (2.14), each natural sloshing frequency $\sigma _{s,2m-1},\, m=1,\ldots ,N$ determines the corresponding eigenvector $\boldsymbol {a}^{(2m-1)}$ whose scalar elements can be written down as

(2.18)\begin{equation} a_i^{(2m-1)}=\frac{\tanh({\rm \pi} (2i-1)h)}{(2i-1)(\sigma_{2i-1}^2/\sigma_{s,2m-1}^2-1)}, \quad i=1,\ldots,N. \end{equation}

When normalising these eigenvectors by $\|\boldsymbol {a}^{(2m-1)}\|=\sqrt {\sum _{i=1}^N (a_i^{(2m-1)})^2}$ (the sum converges as $N\to \infty$), one can introduce the orthonormal eigenbasis

(2.19)\begin{align} {\boldsymbol q}^{(m)}= \frac{\boldsymbol{a}^{(2m-1)}}{\|\boldsymbol{a}^{(2m-1)}\|}=(q_1^{(m)}, q_2^{(m)},\ldots, q_N^{(m)})=(q_{1m}, q_{2m},\ldots,q_{Nm}),\quad m=1,\ldots,N \end{align}

so that the orthogonal matrices $Q=\{q_{im}\}$ and $Q^T$ ($Q^TQ=I$) diagonalise the mass-matrix ${\mathcal {M}}$, i.e.

(2.20)\begin{equation} Q^T{\mathcal{M}}Q=\text{diag} \left(\frac{1}{\kappa_{s,2k-1}}\right), \quad \text{where}\ \kappa_{s,2k-1}=\frac{\sigma^2_{s,2k-1}}{\bar g}. \end{equation}

Employing the eigenvectors (2.19) in (2.11) introduces

(2.21)\begin{equation} \varphi_{s,2m-1}(y,z)=\sum_{i=1}^N q_{im} f_{2i-1}(y) Z_{2i-1}(z),\quad N\to\infty, \end{equation}

which defines the corresponding (orthogonal) antisymmetric non-Stokes standing wave profiles

(2.22)\begin{equation} f_{s,2m-1}(y)=\left.\frac{\partial\varphi_{s,2m-1}}{\partial z}\right|_{z=0}=\sum_{i=1}^N q_{im} f_{2i-1}(y),\quad N\to \infty. \end{equation}

Specifically, the mean square root integral over $f_{2m-1}(y)$ and $f_{s,2m-1}(y)$ are equal, i.e. $\|\,f_{2m-1}\|= \|\,f_{s,2m-1}\|=1/\sqrt {2}$. The standing antisymmetric Stokes wave profiles $f_{2m-1}(y)$ can be restored from $f_{s,2m-1}(y)$ by using $Q$ as follows:

(2.23)\begin{equation} f_{2i-1}(y)=\sum_{m=1}^N q_{im} f_{s,2m-1}(y),\quad N\to \infty. \end{equation}

The symmetric natural sloshing modes coincide with the Stokes modes, $\varphi _{s,2m}=\varphi _{2m}$.

The black solid lines in figure 2(b,c) show the Stokes standing wave profiles $z=f_{1}(y)$ and $z=f_{3}(y)$, which are defined by (2.10). These wave profiles do not depend on the non-dimensional parameters $h$ and $M_t/M_l$. The antisymmetric non-Stokes wave profiles $z=f_{s,1}(y)$ and $z=f_{s,3}(y)$ by (2.22) are functions of $h$ and $M_t/M_l$.

Figure 2(b) focuses on the weightless vehicle, $M_t/M_l=0$. The dashed curves $z=f_{s,1}(y)$ and $z=f_{s,3}(y)$ are labelled by the non-dimensional mean liquid depths ${h=0.3, 0.4, 0.7}$ and 1.5. Figure 2(c) assumes that the mean liquid depth $h$ is fixed ($h=0.5$ in these numerical examples) but the wave profiles $z=f_{s,1}(y)$ and $z=f_{s,3}(y)$ are drawn for $M_t/M_l=0.0, 0.5, 1.0$ and 3.0 (the values are used as the tags).

Comparisons in figure 2(b,c) demonstrate that the non-Stokes natural sloshing modes are increasingly affected by the vehicle sway motions with decreasing $h$ and $M_t/M_l$. It is most clearly seen for the lowest sloshing mode $z=f_{s,1}(y)$. An interesting feature of the non-Stokes standing wave profiles $z=f_{s,1}(y)$ and $z=f_{s,3}(y)$ is that, in contrast to the Stokes wave modes, the high spot point is situated away from the wall. Similar linear ‘high-spot’ results for the Stokes natural sloshing modes are well known for the ice-fishing problem and tanks with non-vertical walls (Kulczycki & Kuznetsov Reference Kulczycki and Kuznetsov2009, Reference Kulczycki and Kuznetsov2011). These are not connected with the flip-through phenomenon, which is of strongly nonlinear nature. As we can see, the coupling also causes the high spot point away from the wall for rectangular tanks with upright walls.

2.4.1. Two equivalent linear modal equations

Let us take the well known linear modal system from § 5.4.2 by Faltinsen & Timokha (Reference Faltinsen and Timokha2009):

(2.46a,b)\begin{equation} \ddot \beta_{2m-1} +\sigma_{2m-1}^2 \beta_{2m-1}=\kappa_{2m-1} \lambda_m \ddot\eta_2;\quad \ddot \beta_{2m} +\sigma_{2m}^2b_{2m}=0,\quad m=1,\ldots,N. \end{equation}

Because the sway motion $\eta _2(t)$ is unknown, using (2.46a,b) requires the Newton law (2.6), which should be rewritten in the form

(2.47)\begin{equation} \ddot\eta_2(t)=\bar{\mathcal{F}}_{ext}(t) +\frac 12 K \sum_{m=1}^N \lambda_m \ddot\beta_{2m-1}(t). \end{equation}

The system of linear ordinary differential equations (2.46a,b), (2.47) describes the linear coupled dynamics in terms of the generalised coordinates $\eta _2$ and $\beta _{2m-1}$. The even hydrodynamic coordinates $\beta _{2m}$ do not effect the coupling and, therefore, can be excluded from the linear analysis.

Alternatively, one can focus on the linearised modal equations (2.31),

(2.48a,b)\begin{align} \ddot b_{2m-1} +\sigma_{s,2m-1}^2b_{2m-1}=P_{2m-1}\bar{\mathcal{F}}_{ext};\quad \ddot b_{2m} +\sigma_{s,2m}^2b_{2m}=0,\quad m=1,\ldots,N. \end{align}

These equations do not contain $\eta _2(t)$. They describe the internal wave motions, not the tank motions. However, when solving (2.48a,b) and substituting $b_{2m-1}(t)$ into (2.43), we get an explicit expression for $\ddot \eta _2(t)$.

The systems of ordinary differential equations (2.46a,b) + (2.47) and (2.48a,b) + (2.43) imply two equivalent linear modal systems, which describe the coupled dynamics. An advantage of (2.48a,b) + (2.43) is that it concentrates on sloshing but the associated tank motions by $\eta _2(t)$ become automatically derivable (known) if we know the hydrodynamic generalised coordinates $b_{2i-1}(t)$. On the other hand, derivations of the linear modal equations (2.48a,b) require knowledge of the non-Stokes sloshing frequencies $\sigma _{s,2i-1}$ and matrix $Q$ while the hydrodynamic coefficients in (2.46a,b) + (2.47) can be computed by using rather simple analytical formulae.

2.4.2. Steady-state wave motions

The steady-state linear analysis suggests the small-amplitude harmonic force

(2.49)\begin{equation} \bar{\mathcal{F}}_{ext}(t)=\sigma^2f_0\cos\sigma t,\quad 0\not=f_0=O(\epsilon)\ll 1. \end{equation}

Substituting (2.49) into either (2.46a,b) + (2.47) or/and (2.48a,b) + (2.43) derives the two equivalent periodic (steady-state) solutions in terms of $\beta _{2m-1}$ or/and $b_{2m-1}$,

(2.50a–c)\begin{align} \beta_{2m-1}(t)=\beta_{0,2m-1}\cos\sigma t;\quad b_{2m-1}(t)=b_{0,2m-1}\cos\sigma t;\quad \eta_2(t)=a_t\cos\sigma t, \end{align}

where

(2.51a,b)\begin{equation} \beta_{0,2m-1}={-}\frac{\sigma^2 \kappa_{2m-1}\lambda_m}{\sigma_{2m-1}^2-\sigma^2}a_t;\quad b_{0,2m-1}=\frac{\sigma^2 P_{2m-1}}{\sigma_{s,2m-1}^2-\sigma^2}f_0. \end{equation}

Furthermore, (2.46a,b) + (2.47) gives the transmission function for the sway amplitude $a_t$:

(2.52)\begin{equation} \frac{a_t}{f_0} ={-} \frac{1}{S(\sigma^2)}, \end{equation}

where the function $S(\sigma ^2)$ was introduced when defining the dispersion equation (2.15). An alternative expression of the dispersion function is also derivable when using the linear modal equations (2.48a,b) + (2.43); the result is

(2.53)\begin{equation} -\frac {1}{S(\sigma^2)} ={-}1 +\frac 12 K \sum_{m=1}^N \frac{\mu_m^2 \kappa_{s,2m-1}}{\bar\sigma_{s,2m-1}^2-1},\quad \bar\sigma_{s,2m-1}=\frac{\sigma_{s,2m-1}}{\sigma}. \end{equation}

The key difference between (2.15) and (2.53) is that the first expression employs the Stokes natural sloshing frequencies $\sigma _{2m-1}$ but the second one is based on the non-Stokes natural sloshing frequencies $\sigma _{s,2m-1}$.

When the forcing frequency $\sigma$ tends to $\sigma _{s,2m-1}$, the second formula in (2.51a,b) and (2.52) ($S(\sigma ^2)\to 0$ as $\sigma ^2\to \sigma _{s,2m-1}^2$) show that both the amplitude $b_{0,2m-1}$ (of the hydrodynamic generalised coordinate $b_{2m-1}$) and the tank amplitude $a_t$ tend to infinity. In other words, $\sigma _{s,2m-1}$ is the resonant frequency for both the liquid and vehicle motions.

When the forcing frequency $\sigma$ tends to the Stokes natural sloshing frequency $\sigma _{2m-1}$, ($S(\sigma ^2)\to \infty$ according to (2.15)), and, therefore, the tank amplitude $a_t$ tends to zero according to (2.52). This limit explains why $\sigma _{2m-1}$ are not the resonance sloshing frequencies anymore, even though the first formula in (2.51a,b) has the denominator $\sigma _{2m-1}^2-\sigma ^2$. Inserting $a_t$ from (2.52) into the formula for $\beta _{0,2m-1}$ in (2.51a,b) and using L’Hôpital's rule proves that $\beta _{0,2m-1}\to {\rm \pi}^2 (2m-1)^2 f_0 /(2K)$ as $\sigma \to \sigma _{2m-1}$, i.e. the sloshing amplitude is finite at the Stokes resonance frequencies unless $M_t/M_l\to \infty$.

Coupling between steady-state sloshing and tank motions introduces both the vehicle and sloshing amplitudes. The vehicle amplitude is associated with $\mbox {max}\, |\eta _2(t)|$, which is, in the linear case, the same as $|a_t|$ (the modulus of the first Fourier harmonic). Because the Stokes natural sloshing modes reach the high spot on the wall, the sloshing amplitude can be associated with max $|\sum \beta _i(t)|$ in terms of the modal representation (2.24) (the maximum wave elevation at the wall). Figure 2(b,c) demonstrates that the high-spot point of $z=f_{s,2m-1}(y)$ is not on the wall and, therefore, analogous direct sum by $b_i(t)$ cannot be used as definition of the sloshing amplitude when adopting (2.29). An alternative definition of the sloshing amplitude could be associated with the $b$-normalised amplitude of the liquid mass centre in the horizontal direction (relative to the $Oyz$ frame)

(2.54)\begin{equation} \max|y_C(t)|, \quad \text{where}\ y_C(t)=\frac{1}{h}\int_{{-}1/2}^{1/2} y\zeta(y,t)\,\textrm{d}y ={-}\frac{1}{2 h}\sum_{m=1}^N \mu_m b_{2m-1}(t), \end{equation}

which is independent of the high-spot position of $z=f_{s,2m-1}(y)$. Physically, $\ddot y_C(t)$ is by the Lukovsky formula (2.5) related to the horizontal hydrodynamic sloshing force. The position of $y_C(t)$ is utilised in phenomenological sloshing models such as the concentrated mass model by Ishikawa et al. (Reference Ishikawa, Kondou, Matsuzaki and Yamamura2016).

Substituting $b_{2m-1}(t)$ by (2.51a,b) derives the linear steady-state liquid-mass motions in the horizontal direction

(2.55)\begin{equation} y_C(t)=a_s\cos\sigma t, \end{equation}

so that $|a_s|$ is the sloshing amplitude in the linear case. The transmission function for $a_s$ takes the form

(2.56)\begin{equation} \frac{a_s}{f_0}={-}\frac{1}{2h} \sum_{m=1}^N \frac{\mu_m^2\kappa_{s,2m-1}}{\bar\sigma_{s,2m-1}^2-1}= \frac{M_t+M_l}{M_l}\left[\frac{1}{S(\sigma^2)}-1\right]. \end{equation}

Comparing (2.56) and (2.52) shows that amplitudes $a_t$ and $a_s$ become infinite and change their sign at $\sigma = \sigma _{s,2m-1}$. As we discussed above, when $\sigma \to \sigma _{2m-1}$, the sway amplitude of the vehicle tends to zero but $a_s/f_0\to -(1+M_t/M_l)$. A particular consequence of the latter limit is that, if $M_t/M_l=O(1)$, $a_s\sim f_0$ at $\sigma =\sigma _{2m-1}$, and, therefore, the resonant steady-state sloshing in the small vicinity of $\sigma _1$ should be rather accurately approximated within the framework of the linear sloshing theory.

The transmission functions $a_s/f_0$ and $a_t/f_0$ by (2.56) and (2.52), respectively, are drawn in figure 3 for $h=0.5$ and $M_t/M_l=1$. Positive values of $a_t/f_0$ and $a_s/f_0$ mean that the liquid mass centre and/or vehicle oscillate in-phase with the external harmonic forcing (2.49) but the negative values imply the out-of-phase oscillation. As usual for the linear harmonically forced oscillator, $a_s/f_0$ (the deep-blue dashed lines), the liquid mass oscillations change the phase when going through the resonance at $\sigma _{s,1}$. However, $a_t/f_0$ (the solid lines) shows that the vehicle oscillations change their phase twice, at the resonance frequency $\sigma _{s,1}$ and at the first Stokes natural sloshing frequency ($\sigma /\sigma _{s,1}= \sigma _1/\sigma _{s,1}=0.874677\ldots$ for the chosen input data) so that the vehicle oscillates in-phase only in the interval $\sigma _1<\sigma <\sigma _{s,1}$. Why $a_t=0$ and $a_s/f_0=-1-M_t/M_l$ at $\sigma =\sigma _1$ is discussed above in the text.

Figure 3.

Transmission functions $a_t/f_0$ (the solid lines) for the tank sway amplitude and $a_s/f_0$ (the deep-blue dashed lines) for the liquid mass amplitude in the horizontal direction by (2.52) and (2.56), respectively. The computations were done with $h=0.5$ and $M_t/M_l=1$. The graphic representation demonstrates that the tank does not move but the liquid mass centre oscillates in the horizontal direction with the amplitude $-1-M_t/M_l$ when $\sigma =\sigma _1$. Positive vertical values of the transmission functions imply the in-phase oscillations with the harmonic external force $\bar {\mathcal {F}}_{ext}(t)$ by (2.49).

2.5.1. The single-dominant modal system

Resonant sloshing with a finite liquid depth and $\sigma$ approaching the lowest natural sloshing frequency is normally well quantified within the framework of the so-called Narimanov–Moiseev asymptotic theory (Narimanov Reference Narimanov1957; Moiseev Reference Moiseev1958; Ockendon & Ockendon Reference Ockendon and Ockendon1973; Faltinsen Reference Faltinsen1974). Faltinsen et al. (Reference Faltinsen, Rognebakke, Lukovsky and Timokha2000) and Faltinsen & Timokha (Reference Faltinsen and Timokha2001) derived the corresponding modal system for the prescribed harmonic tank forcing.

The corresponding derivations for the modal system (2.31) start with the primarily excited generalised coordinate $b_1(t)$, which should, according to the Narimanov–Moiseev theory, possess the lowest asymptotic order $O(\epsilon ^{1/3})$ equipped with the Moiseev-type detuning

(2.57)\begin{equation} \sigma_{s,1}^2/\sigma^2-1= O(\epsilon^{2/3}), \end{equation}

that measures the ‘closeness’ between $\sigma$ and $\sigma _{s,1}$ in terms of $f_0=O(\epsilon )\ll 1$. Assuming $b_1(t)=O(\epsilon ^{1/3})$ and utilising symmetry/antisymmetry of the non-Stokes natural sloshing modes $f_{s,m}(y)$ leads to the following asymptotic relations:

(2.58a–c)\begin{equation} b_1(t)=O(\epsilon^{1/3}),\quad b_{2m}(t)=O(\epsilon^{2/3}),\quad b_{2m+1}(t)=O(\epsilon),\quad m=1,\ldots,N,\ N\to\infty. \end{equation}

Adopting (2.58a–c) and excluding the $o(\epsilon )$-order terms yields the following system of ordinary differential (Narimanov–Moiseev-type modal) equations:

(2.59a) \begin{align} &\ddot b_{1} +\sigma_{s,1}^2 b_{1} + d_2 (\ddot b_1 b_1^2 +\dot b_1^2 b_1) +\sum_{b=1}^N \left( d_{1,b} \left[\ddot b_1 b_{2b} +\dot b_1 \dot b_{2b} \right] + d_{3,b} b_1 \ddot b_{2b} \right)\nonumber\\ &\quad =P_{2m-1}\bar{\mathcal{F}}_{ext}(t), \end{align}

(2.59b) \begin{align} &\ddot b_{2n}+\sigma_{s,2n}^2 b_{2n}+ d_4^{(n)} \ddot b_1 b_1+ d_5^{(n)} \dot b_1\dot b_1=0,\quad n=1,\ldots,N, \end{align}

(2.59c) \begin{align} &\ddot b_{2m-1} +\sigma_{s,2m-1}^2 b_{2m-1} + d_7^{(m)} \ddot b_1 b_1^2 +d_{10}^{(m)} \dot b_1^2 b_1\notag\\ &\qquad +\sum_{b=1}^N \left( d^{(m)}_{6,b} \ddot b_1 b_{2b} +d^{(m)}_{8,b}\ddot b_{2b}b_1 + d_{9,b}^{(m)} \dot b_1 \dot b_{2b} \right)\nonumber\\ &\quad =P_{2m-1}\bar{\mathcal{F}}_{ext}(t),\ \ m=2,\ldots,N. \end{align}

The hydrodynamic coefficients at the nonlinear terms are functions of the mean liquid depth $h$ and the mass ratio $M_t/M_l$. They are computed by the formulae

(2.60a) \begin{align} d_{1,b}&=d_{6,b}^{(1)}=d_{9,b}^{(1)}= d_{1,b}^{oe,1}=t_{b,1}^{eo,1}= \kappa_{s,1}\sum_{k,i=1}^N q_{k1}q_{i1} \frac{d_{2i-1,2b}^{1,2k-1}}{\kappa_{2k-1}}\notag\\ &= \kappa_{s,1} \sum_{k,i=1}^N q_{k1}q_{i1} \frac{t_{2b,2i-1}^{0,2k-1}+t_{2i-1,2b}^{0,2k-1}}{\kappa_{2k-1}}; d_2=d_7^{(1)}=d_{10}^{(1)}= d_{1,1,1}^{ooo,1}=t_{1,1,1}^{ooo,1}\notag\\ &=\kappa_{s,1}\sum_{k,i,j,l=1}^Nq_{k1}q_{i1}q_{j1}q_{l1}\frac{d_{2i-1,2j-1,2l-1}^{2,2k-1}}{\kappa_{2k-1}} = \kappa_{s,1}\sum_{k,i,j,l=1}^Nq_{k1}q_{i1}q_{j1}q_{l1}\frac{t_{2i-1,2j-1,2l-1}^{1,2k-1}}{\kappa_{2k-1}};\notag\\ d_{3,b}&=d_{8,b}^{(1)}= d_{b,1}^{eo,1}=\kappa_{s,1}\sum_{k,i=1}^Nq_{k1}q_{i1} \frac{d_{2b,2i-1}^{1,2k-1}}{\kappa_{2k-1}}; \end{align}

(2.60b)\begin{align} d^{(n)}_4&=d_{1,1}^{1q,2n}=\sum_{i,j=1}^Nq_{i1}q_{j1}d_{2i-1,2j-1}^{1,2n},\quad d^{(n)}_5=t_{1,1}^{0q,2n}=\sum_{i,j=1}^Nq_{i1}q_{j1}t_{2i-1,2j-1}^{0,2n}; \end{align}

(2.60c)\begin{align} &\left. \begin{gathered} d_{6,b}^{(m)} = d_{1,b}^{oe,2m-1}=\kappa_{s,2m-1}\sum_{k,i=1}^Nq_{km}q_{i1} \frac{d_{2i-1,2b}^{1,2k-1}}{\kappa_{2k-1}},\\ d_7^{(m)} =d_{1,1,1}^{ooo,2m-1}=\kappa_{s,2m-1}\sum_{k,i,j,l=1}^Nq_{km}q_{i1}q_{j1}q_{l1}\frac{d_{2i-1,2j-1,2l-1}^{2,2k-1}}{\kappa_{2k-1}},\\ d_{8,b}^{(m)} =d_{b,1}^{eo,2m-1}=\kappa_{s,2m-1}\sum_{k,i=1}^Nq_{km}q_{i1} \frac{d_{2b,2i-1}^{1,2k-1}}{\kappa_{2k-1}},\\ d_{9,b}^{(m)} = t_{b,1}^{eo,2m-1}=\kappa_{s,2m-1}\sum_{k,i=1}^Nq_{km}q_{i1} \frac{t_{2b,2i-1}^{0,2k-1}+t_{2i-1,2b}^{0,2k-1}}{\kappa_{2k-1}},\\ d_{10}^{(m)} =t_{1,1,1}^{ooo,2m-1}=\kappa_{s,2m-1}\sum_{k,i,j,l=1}^Nq_{km}q_{i1}q_{j1}q_{l1}\frac{t_{2i-1,2j-1,2l-1}^{1,2k-1}}{\kappa_{2k-1}}. \end{gathered} \right\} \end{align}

In contrast to the prescribed tank motions when the Narimanov–Moiseev-type modal system (Faltinsen et al. (Reference Faltinsen, Rognebakke, Lukovsky and Timokha2000), (5.24)) nonlinearly couples only three degrees of freedom, $\beta _1(t),\beta _2(t)$ and $\beta _3(t)$, the Narimanov–Moiseev-type modal system (2.59) couples an infinite set of the second-order, $b_{2m}(t)=O(\epsilon ^{2/3})$, and third-order, $b_{2m+1}(t)=O(\epsilon ^{3/3})$, hydrodynamic generalised coordinates.

2.5.2. Steady-state resonance sloshing and its stability

Let us assume (2.49) and $\sigma$ belong to a neighbourhood of the lowest natural sloshing frequency $\sigma _{s,1}$ so that (2.57) is satisfied. Our goal consists of constructing a steady-state wave (periodic) solution of the Narimanov–Moiseev-type modal system (2.59), studying its stability, and analysing the corresponding response curves, which are associated with either the sloshing or tank amplitudes versus the forcing frequency $\sigma$.

The primarily excited generalised coordinate $b_1(t)$ has the lowest asymptotic order and this dominant asymptotic contribution is associated with the first Fourier harmonic, i.e.

(2.61)\begin{equation} b_1(t)=a\cos\sigma t +o(\epsilon^{1/3}),\quad a=O(\epsilon^{1/3}). \end{equation}

Substituting (2.61) into modal equations (2.59b) derives

(2.62)\begin{equation} b_{2n}(t)=a^2 (l_{0,n} +h_{0,n}\cos 2\sigma t) + o(a^3), \end{equation}

where

(2.63)\begin{equation} l_{0,n}=\frac{d_4^{(n)}-d_5^{(n)}}{2\bar\sigma_{s,2n}^2},\quad h_{0,n}=\frac{d_4^{(n)}+d_5^{(n)}}{2(\bar\sigma_{s,2n}^2-4)},\quad \bar\sigma_{s,i}=\frac{\sigma_{s,i}}{\sigma}. \end{equation}

Inserting (2.61) and (2.62) into (2.59a) and gathering all quantities at the first harmonic yields the cubic secular (solvability) condition, which couples the forcing frequency $\sigma$, the forcing amplitude $f_0$ and the dominant sloshing amplitude $a$ as follows:

(2.64)\begin{equation} a \left(\varLambda(\sigma^2) + m_1a^2\right) =P_1 f_0=\varepsilon=O(\epsilon),\quad \varLambda(\sigma^2)= \bar\sigma_{s,1}^2-1, \end{equation}

where $P_1$ is defined in (2.32) and

(2.65)\begin{equation} m_1={-}\tfrac 12 d_2 + \sum_{i=1}^N \left( d_{1,i} \left[{-}l_{0,i} +\tfrac 12 h_{0,i}\right] -2 d_{3,i}h_{0,i}\right). \end{equation}

Furthermore, assuming $a$ is found from the secular condition (2.64) and gathering the corresponding asymptotic terms in (2.59a) up to the $O(\epsilon )$ order derives

(2.66)\begin{equation} b_1(t)=a\cos\sigma t + a^3 \frac{N_{2,1}}{\bar\sigma_{s,1}^2-9} \cos 3\sigma t + o(a^3). \end{equation}

Making the same operation with (2.59c) gives the third-order approximation

(2.67)\begin{align} b_{2m-1}(t)=\frac{P_{2m-1}f_0-N_{1,m}a^3}{\bar\sigma_{s,2m-1}^2-1}\cos\sigma t + a^3 \frac{N_{2,m}}{\bar\sigma_{s,2m-1}^2-9} \cos 3\sigma t + o(a^3),\quad m\ge 2, \end{align}

where

(2.68)\begin{equation} \left. \begin{gathered} N_{1,m}={-}\frac 34 d_7^{(m)}+\frac 1 4 d_{10}^{(m)} +\sum_{i=1}^N \left[ h_{0,i}\left( -\frac 12 d_{6,i}^{(m)}-2d_{8,i}^{(m)}+d_{9,i}^{(m)}\right) -l_{0,i}d_{6,i}^{(m)}\right],\\ N_{2,m}= \frac 14 d_7^{(m)}+\frac 1 4 d_{10}^{(m)}+\sum_{i=1}^N h_{0,i}\left( \frac 12 d_{6,i}^{(m)}+2d_{8,i}^{(m)}+d_{9,i}^{(m)}\right). \end{gathered} \right\} \end{equation}

Specifically, according to the relations between the hydrodynamic coefficients (2.60), $m_1=N_{1,1}$ ($m_1$ is defined in (2.65)) and the secular equation (2.64) implies $a=(P_1f_0-N_{1,1}a^3)/(\bar \sigma _{s,1}^2-1)$. As a consequence, expression (2.67) can uniformly be used for all $m\ge 1$.

Because of the Moiseev detuning (2.57), $\varLambda (\sigma ^2)=O(\epsilon ^{2/3})$ and, therefore, the secular equation (2.64) is asymptotically correct if and only if $m_1=O(1)$. The coefficient $m_1$ is a function of the three non-dimensional parameters $h, K$ and $\bar \sigma _{s,1}=1+\varLambda$. As discussed by Faltinsen, Rognebakke & Timokha (Reference Faltinsen, Rognebakke and Timokha2003), taking $\bar \sigma _{s,1}=1$ in the analytical expression for $m_1$ keeps all the $O(\epsilon )$-order components in the secular equation. The coefficient $m_1$ becomes then only a function of $h$ and $M_t/M_l$. The same logics is acceptable for $N_{1,m}$ and $N_{2,m}$ at $a^3$, which can be assumed independent of $\sigma$.

Stability of the asymptotic solution (2.62), (2.66), (2.67) can be studied by utilising the fast- and slow-time separation and the linear Lyapunov method. The procedure is well described by Faltinsen & Timokha (Reference Faltinsen and Timokha2009, § 9.2.3). Introducing the slow time $\tau = (\sigma \epsilon ^{2/3} t)/2$ and infinitesimal perturbations $\alpha (\tau )$ and $\tilde \alpha (\tau )$ in the lowest-order approximation

(2.69)\begin{equation} b_1= (a+\alpha(\tau))\cos\sigma t + \tilde\alpha(\tau)\sin\sigma t, \end{equation}

inserting (2.69) into the original modal system, gathering the lowest-order terms and the corresponding lowest harmonics, and keeping the linear terms in $\alpha$ and $\tilde \alpha$ leads to the following system of linear differential equations:

(2.70) \begin{equation} \frac {\textrm{d} }{\textrm{d}\tau} \left( \begin{array}{c} \alpha \\ \tilde\alpha \end{array} \right) = \left[ \begin{array}{cc} 0 & -\varLambda - m_1 a^3\\ \varLambda + 3 m_1 a^3 & 0 \end{array} \right] \left( \begin{array}{c} \alpha \\ \tilde\alpha \end{array} \right) = C \left( \begin{array}{c} \alpha \\ \tilde\alpha \end{array} \right). \end{equation}

The fundamental solution of (2.70) is the $\exp (\gamma \tau )$-dependent function where $\gamma$ is one of two eigenvalues of the matrix $C$. These are solutions of the characteristic equation $\gamma ^2+(\varLambda + m_1 a^2) (\varLambda + 3 m_1 a^2)=0$, which means that the steady-state solution is unstable when

(2.71)\begin{equation} (\varLambda + m_1 a^2) (\varLambda + 3m_1 a^2)<0. \end{equation}

2.5.3. Dimension $N$ in (2.59) and secondary resonances

In the limiting case $M_t/M_l\to \infty$ (the vehicle is not affected by sloshing), the infinite-dimensional Narimanov–Moiseev-type modal equations (2.59) couple only three hydrodynamic generalised coordinates, or, in other words, $N=1$, and, in particular, $d_4^{(m)}=d_5^{(m)}=0,\, m\ge 2$ in (2.59b). When $M_t/M_l=O(1)$, all the derived expressions should formally be tested on convergence as $N\to \infty$. However, the tests can be dropped when remembering that we deal with asymptotic approximations within the framework of the Narimanov–Moiseev theory. This means in particular that, if the hydrodynamic coefficients $d_4^{(n)}$ and $d_5^{(n)}$ at nonlinear terms of (2.59b) are comparable or smaller than $O(\epsilon ^{1/3})$, these equations and the corresponding second-order generalised coordinates can be neglected. Specifically, $d_4^{(1)}\sim d_5^{(1)}=O(1)$ and, therefore, conclusions about the asymptotic smallness should involve the ratio $\sqrt {(d_4^{(n)})^2+(d_5^{(n)})^2}/\sqrt {(d_4^{(1)})^2+(d_5^{(1)})^2}$ for $n\ge 2$.

Figure 4(a,b) demonstrates the ratio for $n=1,\ldots ,8$ and $h=0.4$ (panel (a)) and 0.8 (panel b)) with the mass ratios $M_t/M_l=0$, 1, 5, and 10 whose values are employed as labels. Because the realistic forcing amplitudes $\epsilon \sim \varepsilon \sim 10^{-2}$ ($\varepsilon$ is defined in (2.64)), panels (a,b) show that taking $N=4$ in (2.59b) should provide a satisfactory asymptotic approximation of the second-order wave component. Moreover, when $M_t/M_l\ge 5$, one can restrict the asymptotic analysis to the single second-order hydrodynamic coordinate $b_2(t)$ as it happens for the prescribed tank motion in Faltinsen et al. (Reference Faltinsen, Rognebakke, Lukovsky and Timokha2000).

Figure 4.

The relative values (ratios) $\sqrt {(d_4^{(n)})^2+(d_5^{(n)})^2}/\sqrt {(d_4^{(1)})^2+(d_5^{(1)})^2}$ for $n=1,\ldots ,8$, two different mean liquid depths $h$ and four ratios $M_t/M_l$. Panel (a) corresponds to $h=0.4$ but panel (b) is drawn with $h=0.8$. The computed ratios are labelled by the $M_t/M_l$-values. These two panels show that asymptotic approximation of the second-order wave component may be restricted to $b_2, b_4$, $b_6$ and $b_8$ unless the secondary resonance by $b_{2n},\, n\ge 1$ matters. Panel (c) draws curves in the $(h,M_t/M_l)$ plane along which the secondary resonance by $b_{2n},\, n\ge 1$ may occur. It shows that the secondary resonance is only possible for $b_4$ and $b_6$ (the curves are labelled by the indices). Panel (d) demonstrates the pairs $(h,M_t/M_l)$ about which the secondary resonances occur for the third-order hydrodynamic generalised coordinates $b_{2m-1},\, m\ge 2$. For the studied $h$ and $M_t/M_l$, only $b_9(t),\ldots ,b_{15}(t)$ can theoretically be amplified due to the secondary resonance phenomenon.

The higher hydrodynamic generalised coordinates can be affected by the secondary resonance phenomenon. The secondary resonance concept was extensively elaborated and discussed by Faltinsen & Timokha (Reference Faltinsen and Timokha2009, chapter 8) for prescribed tank motions. The secondary (internal) resonance is generated by the higher Fourier harmonics. For the prescribed resonant tank motions, $N=1$ in the corresponding Narimanov–Moiseev-type modal equations and $\sigma \approx \sigma _1$, the secondary resonances are expected when $n\sigma \approx n\sigma _1\approx \sigma _n,\, n\ge 2$. Because the latter secondary resonance condition is based on the natural sloshing spectrum, which, in turn, is a function of $h$, one can theoretically estimate the non-dimensional liquid depth when the secondary resonances occur. According to Faltinsen & Timokha (Reference Faltinsen and Timokha2009, chapter 8), the resonances are expected only in the asymptotic limit $h\to 0$ (in the shallow-water approximation).

The derived Narimanov–Moiseev-type modal equations formally contain an infinite number of $b_{2n}(t),\, n\ge 2$. This derives the secondary resonance condition $\sigma _{s,2n}^2/\sigma _{s,1}^2=4$ with respect to $h$ and $M_t/M_l$ yields two curves in the $(h, M_t/M_l)$-plane, which are depicted in figure 4(c). The curve ‘4’ corresponds to $b_4(t)$ but ‘6’ is responsible for $b_6(t)$. For the hydrodynamic generalised coordinates $b_{2n},\, n\ge 4$, the secondary resonance phenomenon is not possible, at least, in the interval $0.4\le h\le 0.9$. Specifically, the curve ‘6’ in figure 4(c) suggests an unrealistically lightweight vehicle and, therefore, the secondary resonance by the second-order generalised coordinates practically matters only for $b_4(t)$ when $(h,M_t/M_l)$ is close to the upper curve ‘4’. Based on the numerical data in figure 4(a–c), one can restrict the second-order generalised coordinates in (2.59a), (2.59b) to $b_2(t), b_4(t)$, $b_6(t)$ and $b_8(t)$ ($N=3$) for $h$ and $M_t/M_l$ away from the curves ‘4’ and ‘6’ in figure 4(c). Moreover, one can pose $N=1$ when $10\lesssim M_t/M_l$.

The third-order sloshing approximation by the hydrodynamic generalised coordinates $b_{2m-1}(t), m\ge 2$ are governed by (2.59c). These generalised coordinates are ‘driven’, they do not affect the secularity condition (2.64). However, they can potentially be amplified due to the secondary resonance phenomenon, which is associated with zeros of the second denominator $\sigma _{s,2m-1}^2/\sigma _{s,1}^2-9=0$ (the first denominator is never equal to zero as $\sigma \approx \sigma _{s,1}$). Solving $\sigma _{s,2m-1}^2/\sigma _{s,1}^2=9$ with respect to $h$ and $M_t/M_l$ yields the curves in figure 4(d). The curves are labelled by the mode numbers, $2m-1$. Panel (d) demonstrates that only modes $f_{s,2m-1},\, m=5,\ldots ,8$ could be excited due to the secondary resonance. Because these higher wave modes can be significantly affected by viscous damping, their resonant amplification looks unrealistic from a practical point of view.

2.5.4. Soft- and hard-spring type behaviours

Within the framework of the Narimanov–Moiseev theory, the constructed steady-state solution is valid for finite liquid depths, $h=O(1)$, and no secondary resonances occur. Another specific limitation is that the coefficient $m_1=O(1)$ in the secular equation (2.64). When $O(1)=m_1<0$, the nonlinear sloshing response curves in the $(\sigma /\sigma _{s,1},|a|)$-plane should demonstrate the soft-spring type behaviour but $O(1)=m_1>0$ causes the hard-spring type behaviour. If $m_1\approx 0$, the Narimanov–Moiseev theory fails. The limiting case $M_t/M_l\to \infty$ causes the critical depth $h= 0.3368\ldots$, for which $m_1=0$. A detailed analysis of sloshing with this critical depth is given by Faltinsen & Timokha (Reference Faltinsen and Timokha2009), chapter 8.

For the sloshing-affected vehicle, $m_1$ is a function of $h$ and $M_t/M_l$. Figure 5 demonstrates the solid bold curve along which $m_1=0$. The curve divides the $(h,M_t/M_l)$ plane into the two areas; it has a vertical asymptote at $h=0.3368\ldots$. The left-hand area implies the hard-spring type behaviour but the right-hand one means the soft-spring type. These are illustrated by the corresponding schematic response curves in the $(\sigma /\sigma _{s,1},|a|)$ plane.

Figure 5.

The solid bold curve consists of the pairs $(h,M_t/M_l)$ along which $m_1=0$ in the secular equation (2.64). The curve separates the plane into two domains, where the $(\sigma /\sigma _{s,1},|a|)$ response curves (coming from (2.64)) demonstrate the soft-spring ($m_1<0$) and hard-spring ($m_1>0$) type behaviours; when $M_t/M_l\to \infty$, the critical liquid depth tends to $0.3368\ldots$ (Faltinsen & Timokha (Reference Faltinsen and Timokha2009), chapter 8). The embedded pictures schematically depict the soft- and hard-spring type branchings where the solid lines correspond to stable solutions but the dashed lines mark the instability according to (2.71). The turning point $T$ results from intersection with $\gamma _3: \varLambda +3m_1 a^2=0$. The skeleton line $\gamma _0: \varLambda +m_1 a^2=0$ divides the two branches of (2.64). The Narimanov–Moiseev theory fails in a neighbourhood of the dashed red (secondary resonance for the second-order generalised coordinates $b_4$ and $b_6$) and dotted blue (secondary resonance by the third-order generalised coordinates $b_5, b_7,\ldots , b_{15}$ marked in the clockwise direction) lines. Three green squares specify the experimental cases I, II, and III by Rognebakke & Faltinsen (Reference Rognebakke and Faltinsen2003), which are studied in § 3. The experimental cases exhibit realistic values of the structural (+ added mass)/liquid mass ratios. Positions of these ‘green’ experimental points also show that no secondary resonances occur for these experiments and the corresponding amplitude response curves should be of the soft-spring type.

When drawing the solid bold curve in figure 5, we neglected the secondary resonance phenomenon. However, the Narimanov–Moiseev theory fails when these resonances occur and, therefore, one should exclude the pairs $(h,M_t/M_l)$ along which the denominators in (2.63) and (2.68) are zero. These pairs are indicated by the dashed (red) lines for the secondary resonances of the second-order generalised coordinates $b_{2n}$ and the dotted (blue) lines – for the secondary resonances by $b_{2m-1},\, m\ge 2$. The figure also shows three points (in green), which are associated with experimental cases I, II and III on a floating rectangular tank by Rognebakke & Faltinsen (Reference Rognebakke and Faltinsen2003) – the cases will be considered in § 3.

2.5.5. Nonlinear resonance response curves

When working on the linear response curves in figure 3, we defined the vehicle and sloshing amplitudes by selecting the first Fourier harmonics $a_t$ and $a_s$ in the generalised coordinate $\eta _2(t)$ and the horizontal liquid-mass coordinate $y_C(t)$, respectively. In the linear theory, the higher Fourier harmonics do not exist. The nonlinear sloshing theory invertible yields the non-negligible higher harmonics and, therefore, we must return to the original definition by choosing $\max |\eta _2(t)|$ and $\max |y_C(t)|$ as non-dimensional wave amplitudes of vehicle and sloshing, respectively.

Substituting (2.66) and (2.67) into (2.54) gives the Narimanov–Moiseev approximation of the horizontal liquid mass centre,

(2.72)\begin{align} y_C(t)=\underbrace{-(2h)^{{-}1} \left[\mu_1 a +A_0(\sigma) f_0 +A_1 a^3\right]}_{a_s}\cos\sigma t -(2h)^{{-}1} A_2a^3 \cos 3\sigma t +O(a^5), \end{align}

where

(2.73a–c)\begin{align} A_0(\sigma^2)=\sum_{m=2}^N\frac{\kappa_{s,2m-1}\mu_m^2}{\bar\sigma_{s,2m-1}^2-1},\quad A_1={-}\sum_{m=2}^N\frac{\mu_m N_{1,m}}{\bar\sigma_{s,2m-1}^2-1}, \quad A_2=\sum_{m=1}^N\frac{\mu_m N_{2,m}}{\bar\sigma_{s,2m-1}^2-9}, \end{align}

where $A_1$ and $A_2$ can be computed, without loss of generality, at $\sigma =\sigma _{s,1}$ (see, arguments regarding $m_1$ in (2.64)) but $A_0$ is an attribute of the linear theory and, therefore, if we require that the nonlinear steady-state solution (2.72) transforms into the linear one away from the resonance, $A_0$ should be considered as a function of $\sigma ^2$.

Inserting (2.72) into (2.6) gives

(2.74)\begin{align} \eta_2(t)= \underbrace{\left[{-}f_0+\tfrac 12 K(\mu_1a + A_0(\sigma^2)f_0+A_1a^3)\right]}_{a_t}\cos\sigma t +\tfrac 12 K A_2 a^3\cos 3\sigma t +O(a^5), \end{align}

which presents the Narimanov–Moiseev approximation of the steady-state vehicle-sway motion.

In contrast to the linear case, the lowest Fourier harmonics in expressions for $y_C(t)$ and $\eta _2(t)$ are not unique; one should consider the non-zero Fourier component at $\cos 3\sigma t$. As a consequence, computing the response curves should be based on $\max | y_C(t)|$ and $\max | \eta _2(t) |$ instead of $a_s$ and $a_t$. These response curves are drawn and compared with linear transmission functions in figure 6 for input parameters in figure 3. When $h=0.5$ and $M_t/M_l=1$, the nonlinear response curves in the $(\sigma /\sigma _{s,1}, \max | \eta _2(t) |/f_0)$ and $(\sigma /\sigma _{s,1}, \max | y_C(t) |/f_0)$ planes demonstrate the soft-spring type behaviour. It is consistent with expectations in figure 5. The bold solid lines in figure 6 specify the stable steady-state nonlinear sloshing but the magenta solid lines indicate the unstable nonlinear forced sloshing. Condition (2.71) is utilised to detect the stability/instability. As explained in the schematic embedded response curves of figure 5, the instability zone is defined by the skeleton line $\gamma _0: \varLambda (\sigma ^2)+m_1a^2=0$ and the curve $\gamma _3: \varLambda (\sigma ^2)+3 m_1a^2=0$. The latter curve intersects the turning point $T$ on the lower branch of figure 6(b).

Figure 6.

The steady-state amplitude response curves, which characterise vehicle-sway oscillations ($\max | \eta _2(t) |$ by (2.74)) and sloshing (the non-dimensional horizontal amplitude of the mass centre, $\max | y_C(t)|$ by (2.72)) within the framework of the Narimanov–Moiseev asymptotic theory. The soft-spring type behaviour happens for the input date in figure 3, i.e. for $h=0.5$ and $M_t/M_l=1$. The computations are conducted with the non-dimensional forcing amplitude $f_0=0.01$. The nonlinear results depend on $f_0$ but, to easily compare the linear (the dotted blue lines) and nonlinear (the solid lines (stability) and dashed lines (instability)) branching, the amplitudes are normalised by $f_0$. In contrast to the linear case, the tank-sway amplitude is not equal to zero at $\sigma =\sigma _1$ within the framework of the nonlinear theory. However, it has a minimum at the point $M$, which is located slightly to the left of $\sigma _1$.

The dashed deep-blue lines specify the linear transmission functions from figure 3. The linear branching does not depend on the forcing amplitude, but the nonlinear responses are functions of $f_0$; the present computations are made with $f_0=0.01$. The linear theory implies that the tank amplitude is exactly equal to zero at $\sigma =\sigma _1$. However, $A_2\not =0$ and, therefore, the modulus of (2.74) is never equal to zero. The graphs in figure 6 demonstrate, however, a minimum at the point $M$, which is located slightly to the left of $\sigma _1/\sigma _{s,1}$.

To the authors’ best knowledge, the literature does not contain appropriate experimental data to validate the undamped steady-state results in figure 3. However, the constructed steady-state solution will be used in the next section to describe sway of a floating tank in an incident regular wave when the damping due to external wave radiation and the free-surface nonlinearity play an important role. A good agreement with experiments will be shown.

3.1.1. Resonant frequencies

Computations with empty tanks showed that the external flows caused damping of the swaying body, which cannot generally be neglected. However, the undamped analysis of the governing equation (3.6) would be useful for evaluating resonance frequencies, which are associated with the corresponding non-Stokes natural sloshing frequencies $\sigma _{s,2m-1}$ introduced in § 2.2. Indeed, taking ${\mathcal {D}}= {f}_0=0$ in (3.6) and following the analytics in the aforementioned section, we arrive at the dispersion equation (2.15) with the frequency-dependent coefficient $K={K}(\sigma )$ from (3.7a). The lowest resonance frequencies $\sigma _{s,1}$ for the experimental cases I, II, and III are documented in table 1 together with the lowest Stokes natural sloshing frequency $\sigma _1$ by (2.10) and their ratios.

The Stokes natural sloshing frequency $\sigma _1$ is only function of the non-dimensional mean tank liquid depth $h$ and, therefore, it is the same in the cases I and III, where one/two tanks have the same fillings. The lowest resonant frequency $\sigma _{s,1}$ is a complicated function of $h$ and $(M_t+A_{22}(\sigma _{s,1}))/M_l$, which plays the same role as $M_t/M_l$ in § 2. The ratio is presented in the last column of table 1 to show that it decreases with these experimental cases. As a consequence, the frequency ratio $\sigma _1/\sigma _{s,1}$ decreases according to the expectation of the undamped sloshing theory in sloshing-affected vehicles.

3.1.2. Quasi-linear periodic solution

Using the corresponding linear modal steady-state solution from § 2.4.2, which is represented by (2.50a–c), (2.51a,b), adopting the quasi-linear damping terms in (3.6) and gathering all quantities at $\cos \sigma t$ and $\sin \sigma t$, we arrive at the following system of equations:

(3.8)\begin{align} -S(\sigma^2) a_t = {f}_0(\sigma) \cos\alpha;\quad 2\left[\varXi(\sigma)+\varXi_+(\sigma) |a_t| + \varXi_-(\sigma) |a_t|^{{-}1} \right] a_t = {f}_0(\sigma)\sin\alpha \end{align}

with respect to the unknown non-dimensional tank amplitude $a_t$ (the linear solution has only the first Fourier harmonic) and the already-introduced phase lag $\alpha$ between the first Fourier harmonic in $\eta _2(t)$ and the incident wave. Taking the sum of squares and introducing the sloshing amplitude $a_s$ as in § 2.4.2 leads to

(3.9) \begin{align} &a_t^2 \left( S^2(\sigma^2) +4 \left[\varXi(\sigma)+\varXi_+(\sigma) |a_t| + \frac{\varXi_-(\sigma)}{ |a_t|}\right]^2\right) ={f}_0^2(\sigma),\notag\\ &\quad a_s=\frac{a_t}{h\, K(\sigma)}\left[S(\sigma^2)-1\right]. \end{align}

Finding $a_t$ from the first equation and substituting it into the second expression makes it possible to describe the single-harmonic quasi-linear steady-state horizontal motions of the tank and liquid mass centre.

Amplitudes $a_t$ and $a_s$ are finite at the resonance frequency $\sigma _{s,1}$, where $S(\sigma _{s,1}^2)=0$. As in the undamped theory from § 2.4.2, $a_t\to 0$ and the sloshing amplitude is finite, $a_s\to f_0(\sigma _1)/(h\, K(\sigma _1))$, as $\sigma$ tends to the first Stokes natural sloshing frequency $\sigma _1$.

3.2.1. Sloshing-related viscous damping and stability

Equations (3.10) and (3.13) are derived assuming the damping is fully associated with the external flow. However, Rognebakke & Faltinsen (Reference Rognebakke and Faltinsen2003) demonstrated that the sloshing-related viscous damping and wave breaking may matter not only in the transient wave phase. Using the Narimanov–Moiseev asymptotic theory with the single-dominant generalised coordinate $b_1(t)$ in (2.60) makes it possible to link the mean damping with, primarily, damping in the lowest (dominant) equation (2.59a) by incorporating the linear damping term $2\xi _1\dot b_1(t)$, where $\xi _1$ is the (mean) damping ratio.

Going this way and introducing the phase lag $\alpha$ into (2.49) as $\bar {\mathcal {F}}_{ext}(t)=\sigma ^2f_0\cos (\sigma t+\alpha )$ derives the secular equation (2.64) in the form $f_0\cos \alpha =P_1^{-1} a(\varLambda (\sigma ^2)+m_1 a^2)$ as well as the damping-related expression $f_0\cos \alpha =2 a [P_1^{-1}\xi _1]$. Comparing (3.13) with this expression shows that $\varXi _{ext}$ plays the same role as $\varXi _{slosh}=P_1^{-1}\xi _1$ in the Narimanov–Moiseev theory in which the sloshing-related damping is included in the dominant modal equation. Summarising both damping sources means that (3.13) needs to be taken in the form

(3.14)\begin{equation} {f}_0(\sigma)\sin\alpha = 2 a \underbrace{\left(\varXi_{slosh}+ \varXi_{ext}(a,\sigma)\right)}_{\varXi_{damp}(a,\sigma)}. \end{equation}

For each forcing frequency $\sigma$, the system of (3.10), (3.14) governs the lowest-order amplitude parameter in the Narimanov–Moiseev steady-state solution $a$ and the phase lag $\alpha$. Taking the sum of squares gives the following equation:

(3.15)\begin{equation} \frac{{f}_0^2(\sigma)}{a^2} = \left(\frac{\varLambda(\sigma^2)+m_1a^2}{P_1}\right)^2 +4 \, \varXi_{damp}^2(a,\sigma) \end{equation}

with respect to the lowest-order sloshing amplitude $a$.

Accounting for damping in the nonlinear modal equations corrects zones of stable and unstable sloshing. Following Faltinsen & Timokha (Reference Faltinsen and Timokha2017) who show how to separate fast- and slow-time scales in modal systems with linear damping terms derives the sloshing instability condition

(3.16)\begin{equation} (\varLambda(\sigma^2) + m_1 a^2) (\varLambda(\sigma^2) + 3m_1 a^2)<{-}\varXi_{damp}^2(a,\sigma)<0, \end{equation}

which replaces (2.71) for the damped nonlinear sloshing in a rectangular floating tank.

3.2.2. Nonlinear response curves

After solving the secularity (solvability) equation (3.15), the lowest-order amplitude $a$ should be substituted into (2.72), which describes lateral oscillations of the liquid mass centre

(3.17)\begin{equation} y_C(t) =\underbrace{a(B_1+B_3a^2)}_{a_s(a,\sigma)} \cos\sigma t -\frac{A_2}{2h} a^3\cos 3\sigma t +o(a^3), \end{equation}

where

(3.18a,b)\begin{align} B_1(\sigma)={-}\frac{1}{2h} \left(\mu_1+\frac{A_0(\sigma^2)}{P_1}\varLambda(\sigma^2)\right),\quad B_3(\sigma)={-}\frac{1}{2h} \left(A_1+\frac{A_0(\sigma^2)m_1}{P_1}\right). \end{align}

The non-dimensional amplitude $a_t$ by (3.11) appears at the first Fourier harmonic of $\eta _2(t)$. However, the generalised coordinate $\eta _2(t)$ also contains the third Fourier harmonic. Formula (2.74) expresses it for the undamped oscillations with the constant multiplier $K$, which should change to ${K}(3\sigma )$ so that the corresponding formula now takes the form

(3.19)\begin{equation} \eta_2(t)= a_t(a,\sigma) \cos\sigma t +\tfrac 12 {K}(3\sigma) A_2\, a^3\cos 3\sigma t +o(a^3). \end{equation}

Inserting the third Fourier harmonic component of (3.19) into the damping terms of the linear governing equation (3.6) requires using $\varXi (3\sigma )$ instead of $\varXi (\sigma )$, which is rather small according to figure 7(b) and, therefore, this quantity can be neglected. The quantity $\varXi _+(3\sigma )a_t$ has order $O(a^4)$ for the experimental data by Rognebakke & Faltinsen (Reference Rognebakke and Faltinsen2003) and can, therefore, be excluded within the framework of the Narimanov–Moiseev theory. The third quantity is caused by $\textrm {sgn}(\dot \eta _{2b})$ when $\dot \eta _{2b}$ is a single first-harmonic function. If the first Fourier harmonic of $\dot \eta _{2b}$ dominates (as in the present case), $\textrm {sgn}(\dot \eta _{2b})$ does not depend on the third harmonic and the corresponding term in (3.6) only reflects the first Fourier harmonic but the higher Fourier harmonics disappear. The damping of the $3\sigma$ harmonic component can be neglected and (3.19) is the final asymptotic expression for the tank oscillations.